pertaining to \(\Phi (u)=1/\sqrt{u(1-u)}\) . Utilization of generalized statistics as in (REF ) and (REF ) with \(\Phi :(0,1) \rightarrow [0,\infty )\) 
is not new in the literature and in fact has a long history. We give a few examples.
[1]} considers \(\Phi (u)= 1_{[a,1]}(u)u^{-1}\)  with fixed \(0<a<1\) .
His motivation was to measure the relative error (on the region \(\lbrace F_0 \ge a\rbrace )\)  rather than the absolute error. He derives
the limit distributions of \(\sqrt{n}K_{n,\Phi }\)  and \(\sqrt{n}S_{n,\Phi }\)  under \(F=F_0\) , whereas the exact distribution of \(K_{n,\Phi }\)  and \(S_{n,\phi }\)  has been found by
[2]} and [3]}, respectively. Later on [4]} extends his results to \(\Phi (u)=1_{[a,b)}(u) u^{-1}, 0<a<b<1\) . If one is interested in detecting differences in the tails \(\lbrace F_0\le a\rbrace )\)  then
the \(\Phi (u)=1_{(0,a]}u^{-1}\)  can be used. The exact distribution of the pertaining
\(S_{n,\Phi }= \sup _{0<F(x)\le a}\frac{F_n(x)-F(x)}{F(x)}\) 
